Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 335 / 1

Use the outgoing Green function for the three-dimensional Helmholtz equationLet ${\mathbf{n}}^{\prime }$ point out of the obstacle $D$ into the exterior. Apply Green second identity to the field and $G_k$ in a large exterior region. Their Sommerfeld radiation conditions make the outer-sphere contribution vanish. The normal on the exterior region's inner boundary is $-{\mathbf{n}}^{\prime }$, so the Kirchhoff–Helmholtz representation isThe first integral is the incident source field ${\psi }_i$. Therefore the scattered field is the surface term. Since ${\psi }_i$ solves the homogeneous Helmholtz equation within the obstacle, its surface term is zero at exterior observation points; the same representation can be written using only scattered-field traces:The unknown boundary condition determines these traces, but the representation itself does not require choosing that condition.

For large $r=|\mathbf{r}|$, the outgoing kernel hasThus the surface representation of an obstacle far-field pattern, with ${\psi }_s=e^{ikr}{f}_{\infty }/r+O(r^{-2})$, givesConjugating the Herglotz wave function gives $\overline{v_g}({\mathbf{r}}^{\prime })={\int }_{S^2}{e}^{-ik\hat{\mathbf{r}}\cdot {\mathbf{r}}^{\prime }}\overline{g(\hat{\mathbf{r}})}dS$. Interchanging the angular and obstacle-surface integrals yields the Herglotz pairing with an obstacle far field:This contains only the scattered-field boundary traces and the Herglotz function, with the outward-obstacle normal convention fixing the sign.

The boundary identity gives a numerical route to shape recovery from a Herglotz boundary identity. First recover an admissible density $g$ from measured far-field patterns, enforcing the supplied Herglotz pairing with an obstacle far field for available incident directions:Then evaluate its Herglotz wave function throughout a search region and find a smooth positive radial profile satisfyingThe same $h(\theta )$ must work for every azimuth $\varphi$, expressing the assumed axial symmetry. In practice expand $g$ and $h$ in basis functions, enforce these equations at collocation points, and minimize the complex residual together with the far-field-data residual. The exclusion of an interior Dirichlet eigenvalue ensures uniqueness of the interior Dirichlet problem for the trial boundary data. Use regularization of an inverse problem to control measurement error and unstable density components; neither uniqueness nor numerical stability follows merely from writing the residual equations.

For rich incident-direction data, the conjugated pairing constraints form an adjoint far-field integral equation for $g$. For only one fixed incidence, the displayed pairing is a single complex scalar constraint and cannot by itself determine an arbitrary density. Additional data or a joint constrained shape fit is then needed; the trace condition supplies the actual geometric information.